The young Australian defeated the former world number one 4-6, 6-2, 6-4 and will play fourth seed Borna Coric from Croatia on Friday in the last eight. The 32-year-old Murray on Tuesday won his first ATP Tour singles match since career-saving surgery in January with a three-set victory over American Tennys Sandgren.
Having enjoyed a day's rest, he started with intent against De Minaur but failed to get his serve firing consistently and clocked up two double faults as he surrendered his second service game.
But the Briton, number one in the world in 2016 but now a lowly 413, broke back twice and sealed the first set when the Australian seventh seed fired long. The 20-year-old De Minaur burst back in the second set however and Murray looked to be fading physically, the match going into a deciding set.
Murray will play the China Open in Beijing next week as he ramps up his return, but admitted to AFP this week that he does not expect to recapture the form that brought three Grand Slams.